Chapter 13
Fourier Analysis
In addition to their inestimable importance in mathematics and its applications,
Fourier series also serve as the entry point into the wonderful world of Fourier analy-
sis and its wide-ranging extensions and generalizations. An entire industry is devoted to
further developing the theory and enlarging the scope of applications of Fourier–inspired
methods.
New directions in Fourier analysis continue to be discovered and exploited in
a broad range of physical, mathematical, engineering, chemical, biological, financial, and
other systems. In this chapter, we will concentrate on four of the most important variants:
discrete Fourier sums leading to the Fast Fourier Transform (FFT); the modern theory
of wavelets; the Fourier transform; and, finally, its cousin, the Laplace transform. In ad-
dition, more general types of eigenfunction expansions associated with partial differential
equations in higher dimensions will appear in the following chapters.
Modern digital media, such as CD’s, DVD’s and MP3’s, are based on discrete data,
not continuous functions. One typically samples an analog signal at equally spaced time
intervals, and then works exclusively with the resulting discrete (digital) data. The asso-
ciated discrete Fourier representation re-expresses the data in terms of sampled complex
exponentials; it can, in fact, be handled by finite-dimensional vector space methods, and
so, technically, belongs back in the linear algebra portion of this text. However, the insight
gained from the classical continuous Fourier theory proves to be essential in understand-
ing and analyzing its discrete digital counterpart.
An important application of discrete
Fourier sums is in signal and image processing. Basic data compression and noise removal
algorithms are applied to the sample’s discrete Fourier coefficients, acting on the obser-
vation that noise tends to accumulate in the high frequency Fourier modes, while most
important features are concentrated at low frequencies.
The first Section 13.1 develops
the basic Fourier theory in this discrete setting, culminating in the Fast Fourier Transform
(FFT), which produces an efficient numerical algorithm for passing between a signal and
its discrete Fourier coefficients.
One of the inherent limitations of classical Fourier methods, both continuous and
discrete, is that they are not well adapted to localized data.
(In physics, this lack of
localization is the basis of the Heisenberg Uncertainty Principle.) As a result, Fourier-based
signal processing algorithms tend to be inaccurate and/or inefficient when confronting
highly localized signals or images. In the second section, we introduce the modern theory
of wavelets, which is a recent extension of Fourier analysis that more naturally incorporates
multiple scales and localization.
Wavelets are playing an increasingly dominant role in
many modern applications; for instance, the new JPEG digital image compression format
is based on wavelets, as are the computerized FBI fingerprint data used in law enforcement
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Peter J. Olver